MEDB 5502, Module 13, Bayesian Statistics
Topics to be covered
What you will learn
Classical statistics
Bayesian statistics
Prior distribution
Likelihood
Posterior distribution
Applications of Bayesian statistics
Classical statistics, 1 of 3
\(H_0:\ \pi_1=\pi_2\)
\(H_1:\ \pi_1 \ne \pi_2\)
\(\pi_1\)
,
\(pi_2\)
are fixed constants
Classical statistics, 2 of 3
\(T=\frac{\hat p_1-\hat p_2}{s.e.}\)
p-value =
\(P[Z > T]\)
probability of sample results or more extreme
p-value is not
\(P[H_0]\)
Classical statistics, 3 of 3
95% confidence interval
\(\hat p_1-\hat p_2 \pm Z_{\alpha/2}s.e.\)
Range of plausible values
Probability statements not possible.
Break #1
What you have learned
Classical statistics
What’s coming next
Bayesian statistics
A simple example of Bayesian data analysis.
ECMO study
Treatment versus control, mortality endpoint
Treatment: 28 of 29 babies survived
Control: 6 of 10 babies survived
Source: Jim Albert in the Journal of Statistics Education (1995, vol. 3 no. 3).
Wikipedia introduction
P(H|E) = P(E|H) P(H) / P(E)
H = hypothesis
E = evidence
P(H) = prior
P(E|H) = likelihood
P(H|E) = posterior
Prior distribution
Degree of belief
Based on previous studies
Subjective opinion (!?!)
Examples of subjective opinions
Simpler is better
Be cautious about subgroup analysis
Biological mechanism adds evidence
Flat or non-informative prior
Break #2
What you have learned
Bayesian statistics
What’s coming next
Prior distribution
Lay out the parameters
Place half the probability on the diagonal
Difuse prior
Difuse prior
Break #3
What you have learned
Prior distribution
What’s coming next
Likelihood
Likelihood
Break #4
What you have learned
Likelihood
What’s coming next
Posterior distribution
Multiply
Standardize
Main diagonal of posterior probabilities
Upper triangle of posterior probabilities
Break #5
What you have learned
Posterior distribution
What’s coming next
Applications of Bayesian statistics
Applications of Bayesian statistics
Incorporate previous research
Controls from earlier studies
Random coefficient models
Missing data
Non-standard measures
Ranking the best
Summary
What you have learned
Classical statistics
Bayesian statistics
Prior distribution
Likelihood
Posterior distribution
Posterior distribution
Applications of Bayesian statistics